How to Write a Spelling Corrector

In the past week, two friends (Dean and Bill) independently told me
they were amazed at how Google does spelling correction so well and
quickly. Type in a search like [speling] and Google
comes back in 0.1 seconds or so with Did you mean:
spelling.
(Yahoo and Microsoft are similar.)
What surprised me is that I thought Dean and Bill, being highly
accomplished engineers and mathematicians, would have good intuitions
about statistical language processing problems such as spelling
correction. But they didn't, and come to think of it, there's no
reason they should: it was my expectations that were faulty, not their knowledge.

I figured they and many others could benefit from an explanation. The
full details of an industrial-strength spell corrector are quite complex (you
con read a little about it here or here).
What I wanted to do here is to develop, in less than a page of code, a toy
spelling corrector that achieves 80 or 90% accuracy at a processing
speed of at least 10 words per second.

So here, in 21 lines of Python 2.5 code, is the complete spelling
corrector:

The version of edits1 shown here is a variation on one proposed by Darius Bacon; I think
this is clearer than the version I originally had. Darius also fixed
a bug in the function correct.

How It Works: Some Probability Theory

How does it work? First, a little theory. Given a word, we are
trying to choose the most likely spelling correction for that word
(the "correction" may be the original word itself). There is no way to
know for sure (for example, should "lates" be corrected to "late" or
"latest"?), which suggests we use probabilities. We will say that we
are trying to find the correction c, out of all possible
corrections, that maximizes the probability of c given the
original word w:

Since P(w) is the same for every possible c, we can ignore it, giving:

argmaxc P(w|c) P(c)

There are three parts of this expression. From right to left, we have:

P(c), the probability
that a proposed correction c stands on its own. This is called the language model:
think of it as answering the question "how likely is c to appear in an English text?" So
P("the") would have a relatively high probability, while P("zxzxzxzyyy") would be near zero.

P(w|c), the probability that w would be typed in a text when the
author meant c. This is the error model: think of it as answering "how likely
is it that the author would type w by mistake when c was intended?"

argmaxc, the control mechanism, which says to enumerate all feasible
values of c, and then choose the one that gives the best combined probability score.

One obvious question is: why take a simple expression like P(c|w) and replace
it with a more complex expression involving two models rather than one? The answer is that
P(c|w) is already conflating two factors, and it is
easier to separate the two out and deal with them explicitly. Consider the misspelled word
w="thew" and the two candidate corrections c="the" and c="thaw".
Which has a higher P(c|w)? Well, "thaw" seems good because the only change
is "a" to "e", which is a small change. On the other hand, "the" seems good because "the" is a very
common word, and perhaps the typist's finger slipped off the "e" onto the "w". The point is that to
estimate P(c|w) we have to consider both the probability of c and the
probability of the change from c to w anyway, so it is cleaner to formally separate the
two factors.

Now we are ready to show how the program works. First
P(c). We will read a big text file, big.txt, which consists of about a million words.
The file is a concatenation of several public domain books from Project Gutenberg
and lists of most frequent words from Wiktionary
and the British
National Corpus.
(On the plane home when I was writing the first version of the code
all I had was a collection of Sherlock Holmes stories that
happened to be on my laptop; I added the other sources later and stopped
adding texts when they stopped helping, as we shall
see in the Evaluation section.)

We then extract the individual words from the file (using the
function words, which converts everything to lowercase, so
that "the" and "The" will be the same and then defines a word as a
sequence of alphabetic characters, so "don't" will be seen as the
two words "don" and "t"). Next we train a probability model, which
is a fancy way of saying we count how many times each word occurs,
using the function train. It looks like this:

At this point, NWORDS[w] holds a count of how many times the
word w has been seen. There is one complication: novel
words. What happens with a perfectly good word of English that
wasn't seen in our training data? It would be bad form to say the
probability of a word is zero just because we haven't seen it yet.
There are several standard approaches to this problem; we take the
easiest one, which is to treat novel words as if we had seen them
once. This general process is called smoothing, because we are smoothing
over the parts of the probability distribution that would have been
zero, bumping them up to the smallest possible count. This is
achieved through the class collections.defaultdict, which is like a
regular Python dict (what other languages call hash tables) except
that we can specify the default value of any key; here we use 1.

Now let's look at the problem of enumerating the possible corrections c
of a given word w. It is common to talk of the edit distance
between two words: the number of edits it would take to turn one into the other.
An edit can be a deletion (remove one letter), a transposition (swap adjacent letters),
an alteration (change one letter to another) or an insertion (add a letter). Here's a function
that returns a set of all words c that are one edit away from w:

This can be a big set. For a word of length n, there will
be n deletions, n-1 transpositions, 26n
alterations, and 26(n+1) insertions, for a total of
54n+25 (of which a few are typically duplicates). For example,
len(edits1('something')) -- that is, the number of elements in the
result of edits1('something') -- is 494.

The literature on spelling correction claims that 80 to 95% of
spelling errors are an edit distance of 1 from the target. As we
shall see shortly, I put together a development corpus of 270 spelling
errors, and found that only 76% of them have edit distance 1. Perhaps
the examples I found are harder than typical errors. Anyway, I thought
this was not good enough, so we'll need to consider edit distance 2.
That's easy: just apply edits1 to all the results of
edits1:

def edits2(word):
return set(e2 for e1 in edits1(word) for e2 in edits1(e1))

This is easy to write, but we're starting to get into some serious
computation: len(edits2('something')) is 114,324. However, we do get
good coverage: of the 270 test cases, only 3 have an edit distance
greater than 2. That is, edits2 will cover 98.9% of the cases; that's
good enough for me. Since we aren't going beyond edit distance 2, we
can do a small optimization: only keep the candidates that are
actually known words. We still have to consider all the possibilities,
but we don't have to build up a big set of them. The function
known_edits2 does this:

def known_edits2(word):
return set(e2 for e1 in edits1(word) for e2 in edits1(e1) if e2 in NWORDS)

Now, for example, known_edits2('something') is a set of just 4 words:
{'smoothing', 'seething', 'something', 'soothing'}, rather than the set of
114,324 words generated by edits2. That speeds things up by about 10%.

Now the only part left is the error model,
P(w|c). Here's where I ran into difficulty. Sitting on
the plane, with no internet connection, I was stymied: I had no
training data to build a model of spelling errors. I had some
intuitions: mistaking one vowel for another is more probable than
mistaking two consonants; making an error on the first letter of a
word is less probable, etc. But I had no numbers to back that
up. So I
took a shortcut: I defined a trivial model that says all known words
of edit distance 1 are infinitely more probable than known words of
edit distance 2, and infinitely less probable than a known word of
edit distance 0. By "known word" I mean a word that we have seen in
the language model training data -- a word in the dictionary. We can
implement this strategy as follows:

The function correct chooses as the set of candidate words
the set with the shortest edit distance to the original word, as long
as the set has some known words. Once it identifies the candidate set to
consider, it chooses the element with the highest P(c) value, as
estimated by the NWORDS model.

Evaluation

Now it is time to evaluate how well this program does. On the plane I
tried a few examples, and it seemed okay. After my plane landed, I
downloaded Roger Mitton's Birkbeck spelling error
corpus from the Oxford Text Archive. From that I extracted two
test sets of corrections. The first is for development, meaning I get
to look at it while I'm developing the program. The second is a final
test set, meaning I'm not allowed to look at it, nor change my program
after evaluating on it. This practice of having two sets is good
hygiene; it keeps me from fooling myself into thinking I'm doing
better than I am by tuning the program to one specific set of
tests. Here I show an excerpt of the two tests and the function to run
them; to see the complete set of tests (along with the rest of the
program), see the file spell.py.

So on the development set of 270 cases, we get 74% correct in 13
seconds (a rate of 17 Hz), and on the final test set we get 67%
correct (at 15 Hz).

Update: In the original version of this essay I incorrectly
reported a higher score on both test sets, due to a bug. The bug was
subtle, but I should have caught it, and I apologize for misleading
those who read the earlier version. In the original version of
spelltest, I left out the if bias: in the fourth
line of the function (and the default value was bias=0, not
bias=None). I figured that when bias = 0, the statement
NWORDS[target] += bias would have no effect. In fact it does
not change the value of NWORDS[target], but it does have an
effect: it makes (target in NWORDS) true. So in effect the
spelltest routine was cheating by making all the unknown
words known. This was a humbling error, and I have to admit that much
as I like defaultdict for the brevity it adds to programs, I
think I would not have had this bug if I had used regular dicts.

Update 2: defaultdict strikes again. Darius Bacon
pointed out that in the function correct, I had accessed NWORDS[w]. This has the
unfortunate side-effect of adding w to the defaultdict, if w was not already there
(i.e., if it was an unknown word). Then the next time, it would be present, and we would get
the wrong answer. Darius correctly suggested changing to NWORDS.get. (This works because
max(None, i) is i for any integer i.)

In conclusion, I met my goals for brevity, development time, and runtime speed, but not for accuracy.

Future Work

Let's think about how we
could do better. (I've done some more in a separate chapter for a book.) We'll again look at all three factors of
the probability model: (1) P(c);
(2) P(w|c); and (3) argmaxc. We'll look at
examples of what we got wrong. Then we'll
look at some factors beyond the three...

P(c), the language model. We can distinguish two sources
of error in the language model. The more serious is unknown words. In
the development set, there are 15 unknown words, or 5%, and in the
final test set, 43 unknown words or 11%. Here are some examples
of the output of spelltest with verbose=True:

In this output we show the call to correct and the result
(with the NWORDS count for the result in parentheses), and
then the word expected by the test set (again with the count in
parentheses). What this shows is that if you don't know that
'econometric' is a word, you're not going to be able to correct
'economtric'. We could mitigate by adding more text to the training
corpus, but then we also add words that might turn out to be the wrong
answer. Note the last four lines above are inflections of words that
do appear in the dictionary in other forms. So we might want a model
that says it is okay to add '-ed' to a verb or '-s' to a noun.

The second potential source of error in the language model is bad
probabilities: two words appear in the dictionary, but the wrong one
appears more frequently. I must say that I couldn't find cases where
this is the only fault; other problems seem much more serious.

We can simulate how much better we might do with a better language
model by cheating on the tests: pretending that we have seen the
correctly spelled word 1, 10, or more times. This simulates having
more text (and just the right text) in the language model. The
function spelltest has a parameter, bias, that does
this. Here's what happens on the development and final test sets when
we add more bias to the correctly-spelled words:

Bias

Dev.

Test

0

74%

67%

1

74%

70%

10

76%

73%

100

82%

77%

1000

89%

80%

On both test sets we get significant gains, approaching 80-90%.
This suggests that it is possible that if we had a good enough language model we
might get to our accuracy goal. On the other hand, this is probably optimistic,
because as we build a bigger language model we would also introduce words that
are the wrong answer, which this method does not do.

Another way to deal with unknown words is to allow the result of
correct to be a word we have not seen. For example, if the
input is "electroencephalographicallz", a good correction would be to
change the final "z" to an "y", even though
"electroencephalographically" is not in our dictionary. We could
achieve this with a language model based on components of words:
perhaps on syllables or suffixes (such as "-ally"), but it is far
easier to base it on sequences of characters: 2-, 3- and 4-letter
sequences.

P(w|c), the error model. So far, the error model
has been trivial: the smaller the edit distance, the smaller the
error. This causes some problems, as the examples below show. First,
some cases where correct returns a word at edit distance 1
when it should return one at edit distance 2:

The same type of lesson holds: In 'thay', changing an 'a' to an 'e'
should count as a smaller change than changing a 'y' to a 't'. How
much smaller? It must be a least a factor of 2.5 to overcome the prior
probability advantage of 'that' over 'they'.

Clearly we could use a better model of the cost of edits. We could
use our intuition to assign lower costs for doubling letters and
changing a vowel to another vowel (as compared to an arbitrary letter
change), but it seems better to gather data: to get a corpus of
spelling errors, and count how likely it is to make each insertion,
deletion, or alteration, given the surrounding characters. We need a
lot of data to do this well. If we want to look at the change of one
character for another, given a window of two characters on each side,
that's 266, which is over 300 million characters. You'd
want several examples of each, on average, so we need at least a
billion characters of correction data; probably safer with at least 10
billion.

Note there is a connection between the language model and the error model.
The current program has such a simple error model (all edit distance 1 words
before any edit distance 2 words) that it handicaps the language model: we are
afraid to add obscure words to the model, because if one of those obscure words
happens to be edit distance 1 from an input word, then it will be chosen, even if
there is a very common word at edit distance 2. With a better error model we
can be more aggressive about adding obscure words to the dictionary. Here are some
examples where the presence of obscure words in the dictionary hurts us:

The enumeration of possible
corrections, argmaxc. Our program enumerates all corrections within
edit distance 2. In the development set, only 3 words out of 270 are
beyond edit distance 2, but in the final test set, there were 23 out
of 400. Here they are:

We could consider extending the model by allowing a limited set of
edits at edit distance 3. For example, allowing only the insertion of
a vowel next to another vowel, or the replacement of a vowel for
another vowel, or replacing close consonants like "c" to "s" would
handle almost all these cases.

There's actually a fourth (and best) way to improve: change the
interface to correct to look at more context. So far,
correct only looks at one word at a time. It turns out that
in many cases it is difficult to make a decision based only on a
single word. This is most obvious when there is a word that appears
in the dictionary, but the test set says it should be corrected to
another word anyway:

We can't possibly know that correct('where') should be
'were' in at least one case, but should remain 'where' in other cases.
But if the query had been correct('They where going') then it
seems likely that "where" should be corrected to "were".

The context of the surrounding words can help when there are obvious errors,
but two or more good candidate corrections. Consider:

Why should 'thear' be corrected as 'there' rather than 'their'? It is
difficult to tell by the single word alone, but if the query were
correct('There's no there thear') it would be clear.

To build a model that looks at multiple words at a time, we will need a lot of data.
Fortunately, Google has released
a database
of word counts for all sequences up to five words long,
gathered from a corpus of a trillion words.

I believe that a spelling corrector that scores 90% accuracy will
need to use the context of the surrounding words to make a
choice. But we'll leave that for another day...

We could improve our accuracy scores by improving the training
data and the test data. We grabbed a million words of text and assumed
they were all spelled correctly; but it is very likely that the
training data contains several errors. We could try to identify and
fix those. Less daunting a task is to fix the test sets. I noticed
at least three cases where the test set says our program got the wrong
answer, but I believe the program's answer is better than the expected
answer:

Finally, we could improve the implementation by making it much
faster, without changing the results. We could re-implement in a
compiled language rather than an interpreted one. We could have a
lookup table that is specialized to strings rather than Python's
general-purpose dict. We could cache the results of computations so
that we don't have to repeat them multiple times. One word of advice:
before attempting any speed optimizations, profile carefully to see
where the time is actually going.

Errata

Originally my program was 20 lines, but Ivan Peev pointed out that I had used
string.lowercase, which in some locales in some versions of Python, has more
characters than just the a-z I intended. So I added the variable
alphabet to make sure. I could have used string.ascii_lowercase.

Thanks to Jay Liang for pointing out there are only 54n+25 distance 1 edits, not 55n+25 as I originally wrote.

Thanks to Dmitriy Ryaboy for pointing out there was a problem with unknown words; this allowed me to
find the NWORDS[target] += bias bug.

Other Computer Languages

After I posted this article, various people wrote versions in
different programming languages. While the purpose of this article
was to show the algorithms, not to highlight Python (and certainly not
to play "code golf" in an attempt to find the shortest program), the
other examples may be interesting for those who like comparing
languages, or for those who want to borrow an implementation in their
desired language: